½ ¨ 7 Hë H À Sae Mulli (The Korean Physical Society), Volume 57, Number 1, 2008¸ 7 Z 4, pp. 32∼35

¿ ƒ

 ½ ¨ 7 Hë  H À Sae Mulli (The Korean Physical Society), Volume 57, Number 1, 2008¸   7 Z 4, pp. 32∼35

:

¿ R <± n É T “ Ô Ý V ê s„ ÆX c l“ Ó ÞÊ Ý ‘ ¤
¹ ÅF I : ¿ R <± n É T “ Ô Ý V ê s„ ÆX c l“ Ó Þ8 ý R  9:4ß Ã Å  P c lß f Ä  T

Ç S Ë{ ¢] k ù8 ý Ä Z Ø9 0] K ¤• ¤ ¿ R <



™

»Š û Ba : @

^∗

² D

Gw n Ø  æÅ Ò@ /† < Ɠ § “ §€ ª œõ & ñ  Ò, Ø  æÅ Ò 380-702 (2008¸   4 Z 4 1{ 9  ~ à Î6 £ §)

p

™ è& ñ ï  r „  ² ú ˜' Ÿ § > = ~ ½ ÓZ O `  ¦ s 6   x # Œ þ j  H] X  s Ö  ©  © œ  ñ Œ •6   xõ  ¿ º   P : þ j  H] X  s Ö  ©  © œ  ñ Œ •6   x_  q 

 r = 9/4“    y Œ •    s f ç — ¸+ þ A_   © œI à º\  ¦ & ñ S X ‰ >  > í ß –Ù þ ¡ .  © œI à º– РÒ'  ì  rC † < Êà º   H[ þ t_  ì  r

Ÿ

í\  ¦ ¹ 1 Ô ? /# Q r = 9/4“    y Œ •    s f ç — ¸+ þ A_  · ú ˜ 9t t  · ú §“ É r $ í | 9 [ þ t`  ¦ µ 1 ß) €Í Ç x .

PACS numbers: 05.50.+q, 05.70.−a, 64.60.Cn, 75.10.Hk Keywords: yŒ• sf痸+þA, ìrC†<Êú H



©

œ„  s ü < e ” > ‰ & ³ © œ_  s K \  e ” # Q" f  © œ ×  æכ ¹ô  Ç % i 

½ +

É`  ¦ ô  Ç s f ç — ¸+ þ A (Ising model)“ É r s  " é ¶\ " f þ j  H] X  s  Ö



©  © œ  ñ Œ •6   xë ß – e ”   H  â Ä º\  K  · ú ˜ 94 R e ”   [1]. t è ß – 64¸   1 l xî ß – à º\ O s  ´ ú §“ É r ” ¸§ 4 \ • ¸ Ô  ¦½ ¨ “ ¦ ¿ º   P : þ j





H] X  s Ö  ©  © œ  ñ Œ •6   x t  t “ ¦ e ”   H s f ç — ¸+ þ A_  K \  ¦

¹ 1

Ôt  3 l wÙ þ ¡Ü ¼ 9, d ” t # Q  © œ l ‘ : r& h “   & ñ ˜ Г   e ” > & h _  0

Au  t • ¸ · ú ˜ 94 R e ” t  · ú § . ‘ : r“ ¦\ " f  H [ j>  þ jœ í– Ð þ

j  H] X  s Ö  ©  © œ  ñ Œ •6   xõ  ¿ º   P : þ j  H] X  s Ö  ©  © œ  ñ Œ •6   x _

 q  9:4“    y Œ •    (square lattice) s f ç — ¸+ þ A_  $ í

| 9

`  ¦  € Œ •  9“ ¦ r • ¸ô  Ç . 9:4_  q  ×  æכ ¹ô  Ç s Ä »  H s 

° ú

כ   H% ƒ\ " f > _  $ í | 9 s  ß ¼>     l  M :ë  Hs   [2]. ‘ : r

“

¦  H  © œ„  s ü < e ” > ‰ & ³ © œ\  @ /ô  Ç ì  rC † < Êà º   H (partition function zeros) s  : r [3]\   „ ½ Ó`  ¦ ¿ º“ ¦ s f ç — ¸+ þ A`  ¦ ƒ  ½ ¨ ô

 Ç .

þ

j  H] X  s Ö  ©  © œ  ñ Œ •6   x (  ½ + Ë[ jl  J

1

)õ  ¿ º   P : þ j  H ]

X

 s Ö  ©  © œ  ñ Œ •6   x (  ½ + Ë[ jl  J

2

)`  ¦ ”    y Œ •    s f ç

—

¸+ þ A“ É r  A _  K x 9 ž Ðm î ß –\  _ K " f & ñ _  ) a .

H = J

1

X

hi,ji

(1 − σ

i

σ

j

) + J

2

X

hi,ki

(1 − σ

i

σ

k

) (1)

d

”

 (1)\ " f σ

i

  H    & h  i\ " f s f ç — ¸+ þ As  | 9  à º e ”   H Û

¼— 2 ; ° ú כÜ ¼– Ð +1õ  −1`  ¦ 2 [½ + É Ã º e ”  . Õ ªo “ ¦ ' Í   P :

½ +

˓ É r — ¸Ž  H þ j  H] X  s Ö  ©[ þ t (nearest neighbors)\  @ /ô  Ç ½ + Ë s

“ ¦ ¿ º   P : ½ + ˓ É r — ¸Ž  H ¿ º   P : þ j  H] X  s Ö  ©[ þ t (next nearest neighbors)\  @ /ô  Ç ½ + Ës  .  6 £ §õ  ° ú  s  \  -t  E\  ¦ & ñ _  €  

E = 2[r X

hi,ji

(1 − σ

i

σ

j

) + X

hi,ki

(1 − σ

i

σ

k

)] (2)

∗E-mail: [email protected]

s

f ç — ¸+ þ A_  K x 9 ž Ðm î ß –`  ¦  A ü < ° ú  s  ç ß –é ß –y  j þ t à º e ” 



.

H = J

2

2 E (3)

#

Œl \ " f B > h  à º r“ É r

r = J

1

J

2

(4)

–

Ð & ñ _ ÷ & 9 þ j  H] X  s Ö  ©  © œ  ñ Œ •6   xõ  ¿ º   P : þ j  H] X  s  Ö



©  © œ  ñ Œ •6   x_  q \  ¦
  · p . d ”  (2)ü < ° ú  s  \  -t \  ¦

&

ñ

_ ô  Ç s Ä »  H r = 9/4_   â Ä º\  \  -t  % ò ¢ ¸  H € ª œ_ 

&

ñ

à º ° ú כë ß –`  ¦ 2 [½ + É Ã º e ” Ü ¼Ù ¼– Ð s f ç — ¸+ þ A_  ì  rC † < Êà º\  ¦



Ò  ¦ M :  Å Ò Ä »6   x l  M :ë  Hs  .

s

f ç — ¸+ þ A_  ì  rC † < Êà º Z  H  6 £ § d ” \  _ K " f & ñ _  ) a



.

Z = X

{σn}

e

^−βH

(5)

#

Œl \ " f ⍠ H k

B

T _  % i à º– Ð & ñ _ ÷ & 9 k

B

  H ^  ¦Þ Ôë ß –  © œ Ã

ºs “ ¦ T   H “ : r• ¸s  . Õ ªo “ ¦ ½ + ˓ É r — ¸Ž  H 0 p xô  Ç  © œI \ 

@

/ô  Ç ½ + Ës  . d ”  (2)\  ¦ s 6   x €   ì  rC † < Êà º\  ¦  6 £ §õ  ° ú   s

 j þ t à º e ”  .

Z =

E

X

max

E=0

Ω(E)(e

^βJ2/2

)

^−E

(6)

#

Œl \ " f Ω(E)  H  © œI à º (number of states)s “ ¦ E

max

  H

\

 -t  | 9  à º e ”   H þ j@ / & ñ à º ° ú כs  .

‘ :

r“ ¦\ " f  H L × 2L  y Œ •   \  ¦ “ ¦ 9  9 [ j– Ð ~ ½ ӆ ¾ Ó

(|   ~ ½ ӆ ¾ Ó)\ " f  H Å Òl & h “    â > › ¸|  (periodic boundary

-32-

¿ ƒ

 ½ ¨ 7 Hë  H À þ j  H] X  s Ö  ©  © œ  ñ Œ •6   xõ  ¿ º   P : þ j  H] X  s Ö  ©  © œ  ñ Œ •6   x_  q · · · – ^ ” 5 p xƒ   -33-

Table 1. The maximum values of energy, E

_max

, of the L×

2L square-lattice Ising models for r = J

1

/J

2

= 9/4 and L = 3 ∼ 10 (with free boundary conditions in L-direction and periodic boundary conditions in 2L-direction).

Lattice size

Emax

3 × 6 270

4 × 8 504

5 × 10 810

6 × 12 1188

7 × 14 1638

8 × 16 2160

9 × 18 2754

10 × 20 3420

conditions)`  ¦  6   x “ ¦ – Ð ~ ½ ӆ ¾ Ó ( ú ª“ É r ~ ½ ӆ ¾ Ó)\ " f  H   Ä

»– Ðî  r  â > › ¸|  (free boundary conditions)`  ¦  6   xô  Ç .

Table 1“ É r s       > \ " f_  E

max

° ú כ`  ¦ ˜ Ð# Œï  r . p ™ è

&

ñ

ï  r „  ² ú ˜' Ÿ § > = (microcanonical transfer matrix) [4–12]

~

½

ÓZ O `  ¦ s 6   x # Œ  © œI à º Ω(E)\  ¦ & ñ S X ‰ >  > í ß – % i  .

\

V\  ¦ [ þ t€   10 × 20  y Œ •    s f ç — ¸+ þ A_  þ j@ /  © œI à º_ 

&

ñ

S X ‰ô  Ç ° ú כ“ É r  A _  & ñ à º– Ð Å Ò# Q”   .

Ω(E = 2438) = 68750466271198949489691428833 11187029951792078996137747360 (7) s

 à º  H @ /| Ä Ì 6.875 × 10

⁵⁷

\  K { © œ   H …  ;ë  H† < Æ& h Ü ¼– Ð  H Ã

ºs  .  © œI à º Ω(E)\   ƒ  – ÐÕ ª\  ¦ 2 [ €   & ñ S X ‰ô  Ç  ' pà Ô

–

Ðx  ° ú כ[ þ t`  ¦ % 3 `  ¦ à º e ”  .

S(E) = ln Ω(E) (8)

Fig. 1“ É r 10 × 20  y Œ •    s f ç — ¸+ þ A_  & ñ S X ‰ô  Ç  ' pà Ԗ Ð x

 ° ú כ (é ß –0 A: k

B

)[ þ t`  ¦ ˜ Ð# Œï  r .  ' pà Ԗ Ðx   H E = 0õ  E = E

max

\ " f þ j™ è° ú כ 0.693`  ¦ t  9, L = 10_   â Ä º E = 2438\ " f þ j@ /° ú כ 133.2`  ¦ ° ú   H .

&

ñ

S X ‰ô  Ç  © œI à º ° ú כ[ þ t`  ¦ · ú ˜€   d ”  (6)Ü ¼– РÒ'  Z = 0_ 

›

¸| `  ¦ ë ß –7 á ¤   H ì  rC † < Êà º   H[ þ t`  ¦ ½ ¨½ + É Ã º e ”  . d ”  (6)Ü ¼

–

РÒ'  ì  rC † < Êà º   H[ þ t“ É r 4 Ÿ ¤™ èà º ¨ î €  \  Z  ~s >  H † d`  ¦ · ú ˜ Ã

º e ”  .  © œ„  s  ” > rF    H  â Ä º { 9  Ò_  ì  rC † < Êà º   H [

þ

ts  € ª œ_  z  ´Ã º» ¡ ¤`  ¦ – Ðt Ø Ô>   ) a . 7 £ ¤ ì  rC † < Êà º   H [

þ

ts  € ª œ_  z  ´Ã º» ¡ ¤`  ¦ – Ðt Ø Ô  H 0 Au  e ” > & h  (critical point)_  ° ú כ`  ¦   & ñ ô  Ç . Fig. 2  H 4 Ÿ ¤™ èà º a = e

^βJ2/2

¨ î €  

\

" f 3 × 6  y Œ •    s f ç — ¸+ þ A_  ì  rC † < Êà º   H[ þ t`  ¦ ˜ Ð# Œ ï



r . Fig. 2\  ¦ ˜ Ѐ   ¿ º> h_  t  € ª œ_  z  ´Ã º» ¡ ¤   H% ƒ\  e

”

6 £ §`  ¦ · ú ˜ à º e ”  . 7 £ ¤ e ” > & h s  ¿ º> h e ” 6 £ §`  ¦ € Œ ™r  “ ¦ e

”

 .  8  H > \ " f_  ì  rC † < Êà º   H[ þ t_  ì  rŸ í  H Fig. 2ü <

Ä

»   . > _  ß ¼l  & f ” \       H[ þ t_  à º  H Z þ t# Q


0 1000 E 2000 3000

0 50 100 150

S(E)

Fig. 1. Exact entropy S(E) = ln Ω(E) (in unit of k

B

) as a function of energy E of the 10 × 20 square-lattice Ising model for r = 9/4. Here, Ω(E) denotes the exact integer values for the number of states. The states with E = 0 correspond to the ferromagnetic ground states while the states with E = E

max

= 3420 to the antiferromagnetic ground states.

-1.2 -0.4 Re(a) 0.4 1.2

-1.2 -0.4 0.4 1.2

Im(a)

Fig. 2. Exact partition function zeros in the complex a = e

^βJ2/2

plane of the 3 × 6 square-lattice Ising model for r = 9/4. The number of the zeros is 270 in the figure.

The distributions of the partition function zeros of the model on larger lattices show similar patterns to those in the figure. As the lattice size becomes larger, the number of the zeros increases. For example, the 10 × 20 square-lattice Ising model has 3420 zeros.

9 € ª œ_  z  ´Ã º» ¡ ¤   H% ƒ\  e ”   H   H[ þ t“ É r € ª œ_  z  ´Ã º» ¡ ¤\  ] X   H K

 ç ß – .

€ ª

œ_  z  ´Ã º» ¡ ¤\   © œ  î  r   H`  ¦ ' Í   P :   Hs  “ ¦ Â

ÒØ Ô 9 a

1

s  “ ¦ ³ ðr ô  Ç . L × 2L  y Œ •    s f ç — ¸+ þ A

\

" f > _  ß ¼l  Ls  & f ” \     ' Í   P :   H_  z  ´Ã ºÂ Ò

-34- ô  Dz D GÓ ü to † < Æ rt  “D hÓ ü to ”, Volume 57, Number 1, 2008¸   7 Z 4

Table 2. The finite-size values of the first zeros a

^I₁

(L) and the thermal scaling exponents y

_t^I

(L) for the critical point I of the L × 2L square-lattice Ising model with r = 9/4. The last row indicates the extrapolated values in the limit L → ∞.

L a^I1

(L)

y^It

(L)

3 0.794744 + 0.1069707i 0.773563 4 0.765943 + 0.0856282i 0.853477 5 0.749927 + 0.0707793i 0.881463 6 0.740286 + 0.0602714i 0.891492 7 0.734058 + 0.0525326i 0.896617 8 0.729797 + 0.0466049i 0.900800 9 0.726743 + 0.0419135i 0.904944 10 0.724473 + 0.0381018i

∞

0.7106(4) + 0.0002(7)i 0.99(7)

Table 3. The finite-size values of the first zeros a

^II₁

(L) and the thermal scaling exponents y

_t^II

(L) for the critical point II of the L × 2L square-lattice Ising model with r = 9/4.

L a^II1

(L)

yt^II

(L) 3 1.071839 + 0.0417096i 1.106788 4 1.070832 + 0.0303358i 1.076654 5 1.069820 + 0.0238570i 1.059587 6 1.068984 + 0.0196660i 1.048683 7 1.068311 + 0.0167306i 1.041140 8 1.067765 + 0.0145591i 1.035622 9 1.067316 + 0.0128872i 1.031414 10 1.066941 + 0.0115602i

∞

1.0629611(7) − 0.0000002(7)i 1.0001(2)

Re[a

1

(L)]ü < ) ‡Ã ºÂ Ò Im[a

1

(L)]  H » ¡ ¤' ‘  Z O g Ë : (scaling law) [4–12]

Re[a

1

(L)] − a

c

∼ L

^−yt

(9) ü

<

Im[a

1

(L)] ∼ L

^−yt

(10)

\

    \ P % i † < Æ& h  F Gô  Ç (thermodynamic limit)\  • ¸² ú ˜ô  Ç



. # Œl \ " f a

c

  H e ” > & h `  ¦ _ p   9 y

t

  H \ P » ¡ ¤' ‘ t à º (thermal scaling exponent)s  . d ”  (9)ü < (10)`  ¦ s 6   x 

#

Œ Ä »ô  Ç> \ " f_  \ P » ¡ ¤' ‘ t à º y

t

(L)`  ¦

y

_t

(L) = − ln{Re[a

1

(L + 1) − a

c

]/Re[a

1

(L) − a

c

]}

ln[(L + 1)/L] (11) ü

<

y

t

(L) = − ln{Im[a

1

(L + 1)]/Im[a

1

(L)]}

ln[(L + 1)/L] (12)

–

Ð & ñ _ ½ + É Ã º e ”  . ‘ : r“ ¦\ " f  À ғ ¦ e ”   H > _   â Ä º% ƒ! 3  e

”

> & h  a

c

\  ¦ — ¸\  ¦ M :\   H d ”  (12)ë ß –`  ¦  6   x½ + É Ã º e ”  .

ì



rC † < Êà º   H[ þ t_  ì  rŸ í (Fig. 2) ¿ º> h_  t \  ¦ ˜ Ð# Œ Å

ғ ¦ e ” Ü ¼Ù ¼– Ð y Œ •y Œ •_  t \ " f € ª œ_  z  ´Ã º» ¡ ¤\   © œ 



î  r   H`  ¦ ¹ 1 Ô è ­ q à º e ”  . y Œ •y Œ •_    H`  ¦ a

^I₁

(L)ü < a

^II₁

(L)– Ð

³

ðr ô  Ç . Table 2ü < 3“ É r s [ þ t   H_  ° ú כõ  » ¡ ¤' ‘ t à º_  ° ú כ

`



¦ ˜ Ð# ŒÅ ғ ¦ e ”  . > _  ß ¼l  & f ” \     ) ‡Ã ºÂ Ò

&

h

& h   Œ • f ” (7 £ ¤ € ª œ_  z  ´Ã º» ¡ ¤\  & h & h  ] X   H† < Ê)`  ¦ " î S X ‰ > 

· ú

˜ à º e ”  .

Table 2ü < 3_   t } Œ • ×  ¦“ É r BST ~ ½ ÓZ O  [13]`  ¦ s 6   x # Œ

\ P

% i † < Æ& h  F Gô  Ç (L → ∞)\ " f % 3 “ É r ° ú כ[ þ ts  . 7 £ ¤ ¿ º e ” > 

&

h

[ þ t_  0 Au \  ¦ ¹ 1 Ԁ Œ ¤Ü ¼ 9 s [ þ t e ” > & h _  » ¡ ¤' ‘ t à º ° ú כ`  ¦





& ñ Ù þ ¡ . ‘ : r“ ¦_    õ – РÒ'  » ¡ ¤' ‘ t à º  H & ñ S X ‰ >  1s 

|

¨

c  כ s  “ ¦ Æ Ò8 £ ¤K  ^  ¦ à º e ” Ü ¼ 9 s  ° ú כ“ É r e ” > & h \ " f q

\ P s  – ÐÕ ªµ 1 Ïí ß – (logarithmic divergence)† < Ê`  ¦ _ p ô  Ç .

' Í

  P : e ” > & h “ É r ì ø Íy © œ $ í (antiferromagnetism) „  s & h  s

“ ¦ ¿ º   P : e ” > & h “ É r y © œ $ í (ferromagnetism) „  s & h  s

 . r = −9/4“    y Œ •    s f ç — ¸+ þ A“ É r r = 9/4“   — ¸+ þ A õ

 ¢ - a„  @ /g A › ' a> \  e ” Ü ¼Ù ¼– Ð ‘ : r“ ¦_    õ   H r = −9/4“  

—

¸+ þ A_    õ s l • ¸  .

Y c

p w Š à U Ø ”  ô

[1] L. Onsager, Phys. Rev. 65, 117 (1944).

[2] J. L. Monroe and S.-Y. Kim, Phys. Rev. E 76, 021123 (2007).

[3] C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952); 410 (1952).

[4] S.-Y. Kim and R. J. Creswick, Phys. Rev. Lett. 81, 2000 (1998).

[5] S.-Y. Kim and R. J. Creswick, Phys. Rev. E 63, 066107 (2001).

[6] S.-Y. Kim, Nucl. Phys. B 637, 409 (2002).

[7] S.-Y. Kim, Phys. Rev. E 70, 016110 (2004).

[8] S.-Y. Kim, Phys. Rev. Lett. 93, 130604 (2004).

[9] S.-Y. Kim, Nucl. Phys. B 705, 504 (2005).

[10] S.-Y. Kim, Phys. Rev. E 71, 017102 (2005).

[11] S.-Y. Kim, Phys. Rev. E 74, 011119 (2006).

[12] S.-Y. Kim, Phys. Lett. A 358, 245 (2006).

[13] W. H. Press, S. A. Teukolsky, W. T. Vetterling,

and B. P. Flannery, Numerical Recipes in Fortran

77, 2nd edition (Cambridge University Press, Cam-

bridge, 1992), p. 104.

¿ ƒ

 ½ ¨ 7 Hë  H À þ j  H] X  s Ö  ©  © œ  ñ Œ •6   xõ  ¿ º   P : þ j  H] X  s Ö  ©  © œ  ñ Œ •6   x_  q · · · – ^ ” 5 p xƒ   -35-

Partition Function Zeros of the Square-Lattice Ising Model with a Ratio of 9:4 between the Nearest-Neighbor and the Next Nearest-Neighbor

Interactions

Seung-Yeon Kim

^∗

School of Liberal Arts and Sciences, Chungju National University, Chungju 380-702 (Received 1 April 2008)

The exact number of states, Ω(E), of the square-lattice Ising model with a ratio of 9:4 between the nearest-neighbor interaction and the next nearest-neighbor interaction, as a function of energy E, is evaluated for the first time by using a microcanonical transfer matrix. Given the number of states, the exact partition function is obtained. The unknown properties of the model are investigated based on the zeros of the partition function. In particular, the critical points and the thermal scaling exponents of the model are obtained for the first time.

PACS numbers: 05.50.+q, 05.70.−a, 64.60.Cn, 75.10.Hk Keywords: Square-lattice Ising model, Partition function zeros

∗E-mail: [email protected]